Partial Ordering Methods in Nonlinear Problems is a book that will appeal to a very narrow constituency. Someone who is already an expert in the field of nonlinear analysis may find it useful, in that it contains a large number of theorems and proofs. Anyone who is not already familiar with this corner of the discipline, however, will not benefit greatly from this book. The text contains virtually no exposition, and the authors do a less-than-adequate job of putting their results into a broader context. This is hardly the sort of book that invites people to explore a new area of mathematics.
The book is littered with mistakes, both typographical and syntactical. Many of the errors are obvious misspellings (such as “Prelininaries,” “sinve,” “thta,” and “eqyations”), which should have been caught by a simple spell-check. Other misspellings are much more serious. For example, something called Thompson’s metric is important to many of the results in the book. The authors manage to spell Thompson’s name in a variety of ways (even within a single sentence); sometimes it appears correctly, sometimes “Tompson,” sometimes “Thomposon.” The quality of the prose – even if one ignores the spelling errors — is poor. Every significant piece of text, almost without exception, is awkward and ill-phrased. The fact that the authors may not be native English speakers does not excuse this situation. More care should have been taken in the preparation of this book, both on the part of the authors and that of the publisher.
In spite of these flaws, the book may be valuable to a small collection of specialists. It does contain a wide variety of results pertaining to cone theory and nonlinear operators, along with applications to integro-differential equations in Banach spaces. Many of these results are rather new, and it is certainly helpful to have them collected in a single volume. Nevertheless, the utility of this book is severely limited by its technical shortcomings.
Christopher Hammond is Assistant Professor of Mathematics at Connecticut College.